A bill that would give some illegal immigrants access to public financial aid for college will be reintroduced in the Assembly this week, state lawmakers said.

That legislation, the Dream Act, failed in the Senate last March and was subsequently left out of the state budget, after passing in the Assembly.

Because the legislation failed in the Senate, it needs to be reintroduced in the Assembly and then voted on again in that house in order to move forward. If it passes again in the Assembly, it would need to also pass in the Senate, where it failed only three months ago, in order to go to New York Gov. Andrew Cuomo’s desk.

The bill’s Assembly sponsor, Assemblyman Francisco Moya, told The Wall Street Journal that the bill would be reintroduced in the Assembly this week. “The bill is alive,” he said. “The Dream Act is not dead.”

Assembly Speaker Sheldon Silver plans to bring the legislation to a vote again before the end of session, after it is reintroduced this week, his spokesman said.

“We will not stand idly by and let the dreams of New York’s immigrant youth dissolve,” Mr. Silver said in a statement on Monday. “We will lead the way and continue fighting to ensure full passage of the DREAM Act, once and for all.”

The legislation faces an uphill battle. Senate Republicans, including the house’s co-leader, Dean Skelos, all voted against it earlier in the session, and the bill failed 30-29 in the Senate.

To no avail, lawmakers who oversee higher-education policy lobbied for the Dream Act’s inclusion in the state budget, negotiated by house leaders and Gov. Cuomo, after it failed in the Senate.

While the bill impacts only a small group of people, many advocates saw it in part as a symbol of how the state treats immigrants. The failure of the Dream Act in the session so far is among a number of frustrations that more left-leaning Democratic activists have cited with Mr. Cuomo in recent months.

Some supporters blamed Sen. Jeff Klein, a Bronx Democrat now up for reelection, for the bill’s failure. He helped bring the bill to a vote in the Senate and voted for it, but many Democrats said he was to blame for not getting the total votes necessary for its passage.